An in-depth look at piercing

in this guide I want to have a more in-depth look at the piercing ability of scrolls.
I will employ a few mathematical formulas and inspect some of the oddities that you might stumble upon when playing creatures with the piercing ability.
Maybe this is not at all new or interesting, but I still hope that some of you enjoy the presentation regardless.
If you find some errors in the data below please notify me and I will try to correct the error.
And now, without further ado, let us start.

1. The setup

Below, in Fig. 1 you can see the one-row setup I assumed for explaining piercing. The creature with piercing attacks from the left side (the
tile with the gun symbol) and damages creatures, structures, or the idol on the right side of the board. The numbers on the right side of the
board stand for enemy creatures/structures or the idol, ordered so that the attacking creature first damages 1, then 2 and so on. Those do not
have to be on the first or second (and so on) tile from the left. This is only the case in Fig. 1.
The calculations below also work with, say only one creature somewhere on the right side of the row and one idol as shown in Fig. 2. In this
case the creature would be no. 1 and the idol no. 2 (there would not be a 3 or 4 in this case) since for piercing it makes no difference if there is
empty space between the targets hit. Only the number of targets is important.

2. The maximum damage

This part is basically just a mathematical representation of how piercing works if you do not care for mathematical formulas you can just skip
this section without missing too much. It is easier to write a formula for this if you employ the so called "floor function". The floor function takes
any real number and gives you the biggest integer that is still smaller than that number.
The floor function is symbolized by a set of special brackets that look like this ⌊ ⌋

Here are a few examples:

One could also argue that for positive real numbers the floor function just cuts away the decimal places of said number.
Piercing deals full attack damage to the first enemy unit hit and then half the damage (rounded down) to the next unit.
The third unit will be dealt half of the damage (rounded down) that was dealt to the second unit and so on.
This way we can describe the maximum possible damage that can be dealt to the first to fourth enemy unit by employing the floor function:

The total maximum damage is therefore :

3. Damage at a specific position

Knowing the total damage is all fair and well, but what if you want to destroy a very specific enemy unit buried deep behind other units?
In that case maybe we start from the fourth place and see how high the damage to units before the fourth one has to be to deal,say, one
damage in the end:

In Fig. 3 we can see that if we have a fully occupied enemy row we need to deal either 2 or 3 damage to the third unit to have the one
damage we need for the fourth unit left over. If you follow the schematic through to the end though you can see that this is possible if your
attacking unit has an attack value between 8 and 15 (of course an attack value of over 15 would give you even more damage in the end).
If there would be only three units in the enemy row just cut the rightmost column in Fig. 3 off and you have the attack values (4 to 7) you
need to deal exactly one damage to the last enemy unit in the row.

This observation can be generalized, if you want to deal a damage of n on the 4th enemy unit you can calculate the
following lower and upper limits of the damage you need to deal to the 3rd to 1st unit beforehand.

This allows you to see if your goal of damaging an enemy unit in the far back of an occupied row is actually achievable, e.g.
if you have four enemy units and want to deal three damage to the last one with piercing you need at least 8x3 = 24 attack
damage for that.

4. Total damage progression

Looking at Eq. 4 in section 3 we can infer from the numbers given that for every additional point of attack value the damage that can be dealt
to the first enemy unit d₁ raises by 1, every two points the damage dealt to the second unit d₂ raises by 1, every four points d₃ raises by 1
and every eight points d₄ raises by 1.
Taking this into account we can look at two examples that should elucidate the repercussions of this rule:

If you would increase the attack value of "Cannonetta" by one, the value would go to 3 up from 2, but your maximum net damage would only
increase by one, too.

If you take a creature with piercing and an attack value of seven and increase that value by one you can now deal up to four more damage
to enemy units since in that case d₁, d₂, d₃ and d₄ are all increased by one. This is a sizeable amount. And which creature has an attack
value of seven and piercing? Right, "Top Reaver Thea"...

For easier access and to show some landmarks I put some attack values and the corresponding maximum damage you can deal with said
value in a fully occupied enemy row in the graph in Fig.4 below. The coloured arrows indicate damage increases larger than one.
The yellowish arrow symbolizes a damage increase by two, the green one by three and the purple one by four.

As you can see in Fig. 4 (or even in Eq.4) the pattern repeats itself after every eight points of attack value. After zero and then every eighth
point it starts with a damage increase of 1, then 2,1,3,1,2,1,4 per point of attack value.

5. Weak enemy units

The other chapters of this post all dealt with some kind of maximum damage you can deal to an enemy unit, but what if said unit has so little
health that you would not need your entire attack value to kill it? In that case you would potentially "waste" some of your potential damage.
Section 4 shows that piercing can increase the total damage you deal by a geater amount than the increase in the attacking creature's actual
attack value. Now if the first enemy creature you attack only has very little health it is possible to actually deal less total damage with piercing
than the attack value of the attacking creature, just like it would be the case with creatures without piercing.
Table 1 below shows some attack values "a" for an attacking creature with piercing and some health values "h" for the first enemy creature
in the row. Assuming the enemy's part of the row is fully occupied (four creatures/structures + idol) and the the other units apart from the first
all have enough health to completely absorb the damage dealt to them, we can now see if for a given a,h combination you could possibly
deal at least the attacking creatures attack value as total damage to the enemy units.
As for the symbols, ✓ means the attack value a can be sufficient to deal at least the attack value in total damage if the first enemy
creature/structure has h health and ✗ means there is no way - even with a fully occupied enemy row - to deal at least the attack value a in
damage if the first enemy creature/structure has h health.

As you can see piercing is only fully effective for very few attack values if the first attacked unit has one health, only some even numbers
qualify for the green checkmark. That those numbers are mostly even is not that suprising if you consider that we established in section 4
that at those numbers large jumps in the potential net damage occur.
If we increase the health of the attacked unit, more and more attack values join the group of effective values until all of them (at least all of
the shown 16) get the checkmark at four health.
There are basically two trends visible in Table 1. First, the higher the attack value the higher the enemy unit's health has to be to get your
money's worth. Second, attack values that are one lower than the numbers divisible by eight, like 7 and 15, are the least cost-efficient in
each "set of eight".
We can infer from this that it is not always advisable to overly buff your creatures with piercing if you want full damage efficiency, or at least
one should avoid certain benchmark figures like 15 with which you would waste damage against a lot of creatures in Scrolls.

A disclaimer at the end. In Scrolls it is oftentimes not important if you waste a bit of potential damage if you get the job done. Sometimes you just want to destroy a certain important creature
and you do not care too much what else happens with the the rest of the damage you apply. Also all the arguements above do not take effect of enchantments or creature abilities into account so please only take
the content of this post for what it is - a look at very specific scenarios that could or could not happen in the game.

This is the end of this little journey through the world of piercing guns and cannons. I hope you enjoyed the post and found something of value in it.